Balanced-Viscosity solutions for multi-rate systems
Authors
- Mielke, Alexander
ORCID: 0000-0002-4583-3888 - Rossi, Riccarda
ORCID: 0000-0002-7808-0261 - Savaré, Giuseppe
ORCID: 0000-0002-0104-4158
2010 Mathematics Subject Classification
- 34D15 47J30 49J40 49J45
Keywords
- Generalized gradient systems, vanishing-viscosity approach, energy-dissipation principle, jump curves
DOI
Abstract
Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients εα and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.
Appeared in
- MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., vol. 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26
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